Rapid theory-guided prototyping of ductile Mg alloys: from binary to multi-component materials

In order to identify a method allowing for a fast solute assessment without lengthy ab initio calculations, we analyze correlations and anti-correlation between the I 1 ?> stacking fault energies ( I 1 ?> SFEs), which were shown to be related to the macroscopic ductility in Mg alloys, and five material parameters of 18 different elemental solutes. Our analysis reveals that the atomic volume V of pure solutes, their electronegativity ν and bulk modulus B are either linearly or logarithmically related to the I 1 ?> SFE. Comparing the impact of solutes with that of yttrium (that increases the ductility in Mg) we propose a single numerical quantity (called yttrium similarity index, YSI) that is based on these inter-relations. Subsequently, we evaluate this new figure of merit for 76 elements from the periodic table of elements in search for solutes reducing the I 1 ?> SFE. Limiting ourselves first to binary Mg alloys, we hardly find any alternative solutes providing similar I 1 SFE ?> reduction as that due to rare-earth (RE) additions. Therefore, we extended our search to ternary Mg alloys. Assuming that the physical properties of solute combinations can be represented by their average values, 2850 solute combinations were checked and 133 solute pairs (not including any RE elements) have been found to have a YSI larger than 0.85. Quantum-mechanical calculations have been subsequently performed for 11 solute pairs with YSIs higher than 0.95 and they were all found to reduce the I 1 SFE ?> in excellent agreement with the predictions based on the YSI.


Introduction
Magnesium alloys are promising structural materials due to their low weight and high specific strength [1]. However, their broader industrial applications are hindered by their low formability at ambient temperatures. The low room-temperature formability is believed to be caused by an insufficient number of active deformation modes (von Mises criterion) and is also partly associated with the highly anisotropic hexagonal close-packed (hcp) structure of Mg alloys.
It is known that the room temperature ductility of pure Mg can be significantly improved by alloying with a small amount of yttrium or other rare-earth (RE) elements [2][3][4][5]. Recent investigations have shown that the enhanced ductility in Mg-Y alloys is accompanied by an increased density of non-basal dislocations which are supposed to nucleate from I 1 stacking faults [5]. This connection between the I 1 stacking faults and non-basal dislocations has been analyzed by Agnew et al [6] and it has been concluded that the reaction is energetically conceivable for a wide range of fault geometries. Our previous study of this phenomenon combined quantummechanical (so-called ab initio) calculations with state-of-the-art microstructure characterization methods and resulted in a scale-bridging connection between the macroscopic ductility and atomic-scale I 1 stacking fault 2. Computational methodology DFT [21,22] calculations of I 1 SFEs were carried out using the projector augmented wave (PAW) method [23] and electronic exchange-correlation effects were described by the generalized gradient approximation [24] as implemented in the Vienna Ab-initio Simulation Package (VASP) [25,26]. A cut-off plane wave energy of 350 eV was used for all calculations of X Mg 15 alloys and elemental solutes in 2 × 2 × 4 supercells. The Brillouin zones were sampled using dense Gamma-point-based 10 × 10 × 6 k-point meshes to ensure the convergence of total energies within 1 meV per atom. In order to keep a similar solute concentration of both solutes in ternary Mg alloys, a 36-atom supercell 3 × 3 × 4 (Mg34-X1-X2) is applied, in which there are 34 Mg atoms, one solute X1 and one solute X2. Solute X1 is positioned at one corner of the supercell, and solute X2 at its center (figure 1). The advantage of doing this is that in this case a configuration is seemingly most possible to keep the symmetry of hcp, since our yttrium similarity index (YSI) is based on hcp Mg alloys. The fact that two different solutes in supercells visualized in figure 1 are maximally separated reflects minimum interactions among different solutes that we expect in case of their dilute and low concentrations. The cut-off energy employed in computation of ternary Mg alloys is the same as in binary Mg alloys, namely 350 eV, and the gamma based k-mesh is 7 × 7 × 6. These parameters guarantee total energy accuracy within 1 meV per atom. The lattice parameters of pure Mg are employed in the calculation of ternary Mg alloys, which is a good approximation as proven by testing.
Using DFT computed total energies, the I 1 SFEs were predicted in the framework of an Ising model. In this model, the total energy E of a sequence of hcp planes in a hcp metal is decomposed into a series where the atomic planes S i interact via interaction energies J i This model has been successfully applied to different metals, such as fcc iron [29]. Hu et al [30] performed fullpotential augmented plane-wave calculations using the WIEN2K code [31] for hcp Ti and hcp Re and demonstrated that the I 1 SFEs from the ANNNI model agree with those from explicit supercell calculations. We note that the use of the ANNNI model is more justified in case of stacking faults with nearly undistorted local atomic geometries (such as the I 1 ) when the SFE can be expressed in terms of interlayer interaction energies, while different approaches are needed for generalized SFE calculations with highly distorted local atomic geometries (see, e.g., [19,20]). Further, relative volumes changes due to alloying are defined as: Here, Mg is taken as the reference ( V Mg 0 ( ) D = and I SFE Mg 0   15 ) and volumes First we analyzed the relation between the I 1 SFE and the volume per atom in X Mg 15 alloys. Figure 2 shows the computed I 1 D SFEs as a function of the relative atomic volumes of these alloys. Generally, the I 1 SFE is lower when the volume of the alloy is larger. This strong anti-correlation between I 1 D SFE( X Mg 15 ) and ΔV ( X Mg 15 ) can be mathematically expressed by a Pearson correlation coefficient r 0.97 = -. Mg-RE alloys have a larger atomic volume than pure Mg, and thus they lower the I 1 SFE with respect to pure Mg. The I 1 D SFE ( X Mg 15 ) values for elements of the lanthanide series are very close to each other, probably because they share a similar electronic structure. Mg-Y deviates from this pronounced trend. When comparing Y and Sc, Mg-Y has a smaller volume than Mg-Sc, but a lower I 1 D SFE ( X Mg 15 ) than Mg-Sc (that is 50% higher). One origin of this difference may lie in their electronic structures. Scandium and yttrium belong to the same group (IIIB group) as the lanthanide series elements, but they are in different periods and, consequently, have different outer electron shells. The electrons in these atomic shells may account for the differences in the I 1 SFE. The relation between the I 1 SFE and the period number will be discussed in detail below.
Almost all Mg-alloys containing non-RE solutes have negative ΔV ( X Mg 15 ) values and positive I 1 D SFE ( X Mg 15 ) values. The only exception is Tl; Mg Tl 15 has a larger volume than Mg and also higher I 1 SFE than Mg. As the non-RE elements are from different groups and periods, they share hardly any similarity in their electronic structures. Therefore the data points of Mg-non-RE alloys are more scattered around the interpolating line. Importantly, we can conclude that the volume per atom in Mg alloys is strongly anti-correlated with its I 1 SFE and the atomic volume of alloys is a reliable indicator of the SFE changes. The question is if we can further connect the volumetric changes in alloys with the atomic volume of pure solutes.
Having the SFE in Mg-alloys related to the atomic volume in these alloys, we further analyze the latter as a function of the atomic volume of elemental solutes. Figure 3 shows the relation between the atomic volume of a solute X and the average atomic volume V of X Mg 15 alloys. It is worth mentioning that DFT methods usually predict the lattice parameters very accurately with only very small deviations from experimental data. Here both volumes are derived from ab initio calculations. The values follow a rather intuitive trend: when a larger atom X is placed into a X Mg 15 supercell, the average atomic volume of the alloy increases. Most of the elements considered follow this trend. The high Pearsons correlation coefficient r = 0.95 indicates a strong correlation between the elemental volume and the alloy volume. Nevertheless, there are exceptions from this rule. For example, elemental zirconium (Zr) is larger than Mg, but the atomic volume of Mg Zr 15 is smaller than the volume of both pure Zr and Mg. Another exception is Tl. The volume of Tl is 30% larger than Mg, but when incorporated into the Mg matrix it changes less than 1% of the average volume of Mg 15 Tl. We assume that strong chemical bonds are formed between Mg atoms and these solutes (e.g. Zr, Tl) and subsequent electron transfers lead to this anomaly. In the non-RE group, Tc, Ru, Re and Os deviate slightly from this linear trend. These four transition metals have approximately the same volume and all significantly reduce the atomic volume of X Mg 15 (which is even smaller than that in Mg 15 Be). The contracted volumes also indicate formation of strong chemical bonds.
3.1.2. I 1 SFEs in pure solutes and Mg alloys Figure 4 shows the relation between the I 1 SFEs in (i) X Mg 15 alloys and (ii) pure solutes X. The SFE of X Mg 15 increases slowly with the SFE of pure X but a clear linear trend is not evident. Still, the increasing trend can be interpreted such that if an I SF 1 is energetically difficult to be generated in a metal X, so it is in the X Mg 15 alloy. The points can be fitted to a linear relation but the corresponding correlation coefficient, r = 0.67, is rather low.
Interestingly, the slope is 0.061 ± 0.015 » 0.0625 = 1/16, which is exactly the atomic concentration of element X in X Mg 15 . However an approximation based on the fractional concentrations is only roughly indicative. Among the studied elements, Re, Co, and Tc possess medium-level SFEs but when added into X Mg 15 supercells,  in the alloys. In the ANNNI model, the I 1 SFE is proportional to the term (E dhcp -E hcp ), i.e., the total energy difference between the phases with hcp and dhcp stacking sequences. When I 1 SFE is negative, E E dhcp hcp < , the alloy is more energetically stable in a dhcp structure than in the hcp one.

I 1 SFEs and the atomic number of solutes
The lanthanide elements have similar affinities and chemical properties which is a result of similar atomic and electronic structures, and they are often visualized in the periodic table of elements as a single element. Therefore we have defined an average I 1 SFE for the considered lanthanide elements. The I 1 SFEs as a function of period are shown in figure 5(a) and (b). The I 1 SFEs of elemental RE and non-RE solutes decrease with increasing periodic number, but the slope levels down for higher periodic numbers. A similar trend is predicted also for the respective X Mg 15 (X = RE) alloys but with a smaller slope. All ten Mg-non-RE alloys considered, which belong to three different periods, are shown as function of group number (see figure 5(c)). The I 1 SFEs of X Mg 15 of these alloying elements increase when moving from group IVB to VIIIB and decrease from group VIIIB to IIIA (see the values for periods 4 and 6). Figure 5(d) shows the I 1 SFE of five RE elements and four non-RE elements in Period 6. From Pr to Os, the I 1 SFE values increase with the atomic number but then they decrease (see the data point for Tl). The increasing rate from Lu to Os is higher compared to that from Pr to Lu.
If we do not collapse all lanthanides into a single data point but visualize them separately, the I 1 SFEs of lanthanides are shown in figure 6. The I 1 SFE values grow as a function of the atomic number within the lanthanide series in case of both elemental solutes and Mg-alloys containing them. The two trends in I 1 SFEs as a function of the atomic number (lanthanides and Mg-alloys) are different and, therefore, they are individually depicted in figures 6(a) and (b). The I SF 1 energies in pure solutes grow in a concave manner as a function of the atomic number (figure 6(a)) while for X Mg 15 alloys, the I 1 SFE values first increase in a convex manner and then increase in a concave manner for higher atomic numbers (figure 6(b)). Fortunately, both trends deviate from linear trends only very weakly and so correlations are not significantly reduced.

I 1 SFEs and electronegativity
The I 1 SFEs computed for different binary Mg alloys as a function of the electronegativity of the individual alloying elements are depicted in figure 7. The data for the electronegativity have been taken for all solutes considered (both RE and others) from [32]. Even though some local trend (e.g. the four elements around 1.6 electronegativity) is seen, globally the values of I 1 SFE in X Mg 15 increase with increasing electronegativity of elemental solutes. The data points seem to be strongly correlated and the corresponding Pearsons correlation coefficient, r = 0.95, is very high. The electronegativity is an excellent linear indicator of SFE changes induced by alloying.
3.1.5. I 1 SFEs and bulk modulus Figure 8 shows the relation between the I SF 1 energies in X Mg 15 alloys and the value of the bulk modulus of solute elements. The bulk moduli are calculated for all metals considered at their ground state structure, i.e. dhcp crystals in the case of Pr and Nd and hcp crystals for all other considered metals. The I SF 1 energies in X Mg 15 alloys clearly grow with the bulk modulus computed for the elemental solutes but the trend visible in figure 8 is not a simple linear function. We propose a logarithmic relation between both quantities, as it seems to fit better and gives a rather high correlation coefficient r = 0.90. In linear fitting, this coefficient is 0.87. The bulk modulus is thus a good logarithmic indicator of I 1 SFE changes upon alloying.

YSI of binary Mg alloys
We have studied five separate dependencies between the I 1 SFEs of Mg alloys on the one hand and various materials characteristics of elemental solutes on the other. We find that three of them possess strong correlations or anti-correlations with the I SF 1 energies, specifically, (i) the atomic volume; (ii) electronegativity; and (iii) the bulk modulus. The corresponding Pearsons correlation coefficients are −0.97, 0.95 and 0.90, respectively. Interestingly, the I SF 1 energies are found to be (anti-) correlated to (i) the volume, i.e., a structural characteristic, (ii) the electronegativity which is an electronic structures property, and (iii) the bulk modulus, one of the elastic parameters. Similar mutual (anti-) correlations were previously found also in other systems. For example, in the case of body-centered-cubic Fe-Ti alloys, their thermodynamic stability (expressed in terms of their formation energies) were found to be strongly related to their density of state at the Fermi level (an electronic structure characteristics). Furthermore, each of these properties (the formation energy and the density of states) were found to be strongly (anti-) correlated with the polycrystalline Youngs modulus of these alloys [33]. Other (anti-) correlations between the thermodynamic stability and elastic parameters were found in face-centered-cubic alloys e.g. Ni-W [34] or Ni nitrides [35]. Apparently, such relations between the atomic-scale electronic structure parameters and macro-scale properties are rather independent on the actual crystal structure and material system.
A comparison of the above discussed parameters is presented in figure 9, where all elements are sorted according to their (a) atomic radius, (b) electronegativity and (c) bulk modulus. From figure 9 it is evident that all elements which reduce the I 1 SFE in Mg (lanthanides, Y, Sc) have (i) large atomic radii of  184 pm (Mg: 145 pm), (ii) electronegativity values close to the that of Mg (1.1-1.3) and (iii) bulk moduli close to Mg, i.e. 32-56 GPa (see our recent paper [8] for further details). A more detailed analysis of these parameters shows that Sc, which only moderately reduces the I 1 SFE in Mg, fulfills all three criteria but its values are a bit away from those of Y and the lanthanides. Additionally, it was revealed that none of the other elements (i.e. others than lanthanides, Y and Sc) fulfills all three criteria.
Having all these above discussed findings we see that there are three critically important parameters of solutes (the atomic volume, electronegativity, and the bulk modulus) that should be considered. As it is rather inconvenient to simultaneously consider multiple criteria, we below attempt to provide a single evaluation factor integrating all three parameters. Due to the fact that we aim at ductility improvements, we use yttrium as a reference below because it is known to improve the ductility of Mg alloys [2,4,5]. Below we thus compare properties of other solutes with those known for Y. Based on the above summarized relations we define an YSI, a new numerical indicator which combines the atomic volume, electronegativity and the bulk modulus. We define the YSI as follows  Cs (117 Å 3 ), the electronegativity of F (3.98), and the bulk modulus of Os (462 GPa). As the reference, the atomic volume, electronegativity and bulk modulus of yttrium are 33 Å 3 , 1.22 and 41 GPa. The YSI parameter defines a specific metric or 'distance' between an element and yttrium in a property space. The larger YSI is, the more similar it is to yttrium. It should be noted that the YSI is introduced as a concentration-independent quantifier because it compares only concentration-independent materials characteristics of elemental solutes with corresponding values that are known for pure yttrium. The fact that most of I SF 1 energies discussed in our study were computed for one specific solute concentration (6.25 at.%) is therefore of only limited importance.
The three selected characteristics of 76 elements from the periodic table of elements are together with their computed YSI values summarized in table 1. For the lanthanide series (from Ce to Lu), the YSI values are very large and close to each other. By using the YSI we can identify the lanthanide elements as the most similar elements to yttrium (see table 2). That matches the above discussed I 1 SFEs which are low for Mg-RE binaries. The non-RE elements with the highest YSI values (0.84-0.86) are Na, Ca, Zr, Tl, and Li. Judging from their YSI values which are very close to Mg (0.84), these non-RE elements do not guarantee a reduction of the I 1 SFE in Mg alloys since our model is based on an approximation of three (anti-)correlations. Calcium was found to decrease the elongation of Mg up to a concentration of 3 wt% [37]. Salahshoor et al performed a more dense set of experiments on samples with 0-4.0 wt% of Ca [38]. They found that Ca could increase the elongation of Mg by 4% for an alloying content of up to » 0.5 wt% but decreases the elongation for higher concentrations. There are, unfortunately, no experimental data for Mg-Na alloys available. The solubility of Na in Mg is very low (see Mg-Na phase diagram), therefore Na is not a promising solute of interest. Zirconium has a too low solubility in Mg, hence it is used as a grain refiner [39]. No data for Mg-Tl alloys are available. Li is a well-known solute that can improve the ductility of Mg [40][41][42][43] by enhanced activity of c a á + ñdislocations [43], which is consistent with our prediction. There are no bulk modulus data available for H, N, O, F, Ga, Tc and In. The bulk modulus term in the YSI determination is then ignored by setting B i = B Y = 41 GPa. However, it was found that these (mostly interstitial which is characterized by a high Pearson correlation coefficient r = 0.93. The exponential character of the trend in figure 10 is partly due to the logarithmic relation between the bulk moduli and the I 1 SFEs. In our definition of the YSI, we use linear functions (concept of distances in a generalized space of materials properties of elements). By using this exponential function we can easily find the lowest values of the YSI parameter that leads to a reduction of the I 1 SFE in Mg alloys (compared with elemental Mg). This critical value is associated with the I 1 SFE of pure Mg (19.47 mJ m 2 -), and the threshold value of YSI is 0.84 1 1 4 ln 2 ( ) » -. In figure 10. Therefore, the elements which can possibly reduce I 1 SFEs are expected to have the YSI values above 0.84 and the I 1 SFE lower than 19.47 mJ m 2 -. For elements with YSI  0.84, their locations in the properties space are shown in figure 11 (some elements with YSI smaller than 0.84 are also shown).
Our findings can be interpreted as sever fundamental limitations in any search for alternative solutes ductilizing magnesium (there is only a single non-RE element, Li, having a YSI  0.84). Consequently, alloydesign strategies should be rather focused on multi-component alloying concepts (e.g., ternary or quaternary Mg-alloys) that can potentially provide a necessary reduction of the I 1 SFE as yttrium by combining more than one solute. In order to pave this way, we below extend our searching space of solutes from binaries to ternaries and demonstrate the performance of our screening and search method.

YSI and I 1 SFE of ternary Mg alloys 3.3.1. Extending the concept of YSI to ternaries
The YSI concept introduced above (see equation (7) is based on comparing selected materials properties of individual solutes to corresponding materials properties in yttrium with the aim to identify individual solutes. In case of Mg-based ternaries containing two different solutes at once, we would like to assign a single numerical evaluator to a pair of solutes, each of them having different elemental materials properties.
In order to extend the YSI concept to multi-component alloys, we propose to use solute-ratio weighted averages of materials parameters of individual solutes, i.e., averages of materials properties of both individual participating solutes in case of two solutes with an equal atomic-percent amount (see the cases discussed below). In the ternary case we then re-write the YSI formula for a single solute (equation (7)) and extend the YSI concept to ternary Mg alloys as follows   and w B as in equation (7). The total number of solutes pairs (containing two different solutes) is 76 76 2 76 2850 -= . Applying equation (9), the YSI values of all 2850 combinations were calculated. Since our aim is to identify solute pair that consists of non-RE elements but is also able to reduce the I 1 SFE like yttrium, the RE elements are excluded from our discussion of results below.
No non-RE candidates possess YSI above 0.98. The statistical data of the matrix are listed in table 3. There are 133 non-RE solute pairs witha YSI larger than 0.85. These solute candidates along with their YSI are listed in table 4 ordered according to their YSI value from the lowest to the highest one. The last 11 solute pairs in table 4 have YSI values larger than 0.95. Except (Sr,Zn), ten of these pairs contain either Ca, Na or Al. Among these ten pairs, five solute pairs contain Ca, three include Na and two contain Al.

Ab initio computed SFEs in Mg-based ternaries
In order to test the reliability of the above predictions of solute pairs based on the YSI, we below determine the I 1 SFEs for 11 promising solute pairs from the ANNNI model. To do so, quantum-mechanical calculations have been performed for supercells with two non-equivalent configurations of two solutes in both hcp and dhcp supercells (shown in figure 1). The solutes are located so as to maximize their mutual distances within the supercells. The chosen arrangements minimize their mutual interactions as expected in case of low and dilute concentrations. Minimum interactions between solute atoms are also in line with our proposed extension of the energy is negative in case of the (Zr,Ca), i.e., the energy of the (Zr,Ca) pair in case of the dhcp-2 supercell is even lower than that in case of the hcp-2 supercell, and that would indicate that the stacking faults would form very easily, in fact spontaneously, if allowed by surrounding crystal matrix in this particular case.
At T = 0 K, the solutes choose their lattice position so as to minimize the energies (both in dhcp and hcp crystals) and the corresponding computed values of I 1 SFE are in table 5 in the 'min.' column (these I 1 SFE values are the smallest values among '1-1', '1-2', '2-1' and '2-2' combinations for each considered solutes pair). At finite temperatures, the solutes have non-zero probabilities to occupy also other positions, not only those that minimize the energy. As an estimate for the finite-temperature values of the I SF 1 energies, we assume an equal probability for all crystal sites and compute an average value of the four I 1 SFEs and list them in table 5 in the column marked 'average'. We expect real values somewhere between the minimum value of the I 1 SFE and the average one. Therefore, we sort the solute pairs so that if both the minimum and averaged values are smaller than 19.47 mJ m 2 -(the value found in the elemental Mg), the corresponding solute pair is considered as 'very promising' with respect to the reduction of the I 1 SFE in Mg ternaries. If only one of these two quantities is smaller than 19.47 mJ m 2 -, the solute combination is considered as 'promising' to reduce I 1 SFE. Lastly, if both quantities are larger than 19.47 mJ m 2 -, the candidate pair is evaluated as 'not promising' to reduce I 1 SFE.
Following the above explained evaluation, eight solute combinations, (Ti,Ca), (Sr,Al), (Tl,Ca), (Sr,Zn), (Hf, Ca), (Zr,Na), (Tl,Na) and (Zr,Ca), are predicted to be very promising with respect to reducing the I 1 SFE in Mg alloys. Three other solute pairs, (Cd,Na), (Ca,Al) and (Cd,Ca), are promising to reduce I 1 SFE of Mg. As seen, no solute pair was found 'not promising', i.e. unable to reduce the I 1 SFE in Mg alloys. That means an excellent agreement between our YSI-based prediction and benchmarking I 1 SFE quantum-mechanical calculations. The SFEs discussed above were computed using lattice constants of pure Mg (a 3.1886 = Å, c a 1.6261 = ). To assess an error introduced by this way of calculating the energies, the lattice constants of one specific solute pair (we choose Mg-(Al,Ca) as both Al and Ca are used as alloying elements in Mg alloys) were fully optimized and found equal to a 3.199 = Å, c a 1.626 = . The subsequent use of the optimized lattice constants in our quantum-mechanical calculations resulted in the SFEs altered by only 0.7-0.8 mJ m 2 and the use of the Mg lattice constant is thus shown to be a good approximation. Nevertheless, it should be noted that our approach based on the YSI does not include more complex thermodynamic aspects of ternary alloys that are related to e.g. ternary phase diagrams where intermetallics and other phases appear and influence, for example, the solubility of solute(s) or thermodynamic stability of ternary alloys. We also note that an alternative solute evaluator was very recently published by Liu et al [44] which considers also electronic-structure characteristics of individual solute pairs. This alternative approach can Table 5. The ab initio computed I 1 SFEs of ternary Mg alloys with solutes consisting of only non-RE elements with YSI larger than 0.95. The I SF 1 energies were computed using the ANNNI model from the four energies of hcp and dhcp supercells depicted in figure 1. They are noted as '1-1', '1-2', '2-1', and '2-2' when, e.g., '1-2' means the I 1 SFE is computed from the energy difference of E dhcp 1 -and E hcp 2 -. The unit of the numbers is mJ m 2 -. The last column contains a verbal evaluation of the potential of listed solute pairs with respect to the reduction of I 1 SFE in Mg-based ternaries. possibly better capture the thermodynamic complexity of solutes in Mg alloys (see also the recent paper by Cui et al [45]). But, in order to do so, it requires quantum-mechanical calculations to be performed for each solute combination and such procedure may not be easily inverted, i.e. identify a solute combination for a given SFE value. In contrast, the YSI was purposely designed to avoid any direct ab initio calculations (albeit it is firmly based on our previous quantum mechanical calculations) and its mathematical backbone allows for an inverse materials design. Another specific aspect is interstitial solutes, that typically compare poorly with yttrium using the YSI, and in case of them we suggest the reader to check, for example, the recently published paper by Zhang et al [46].

Conclusions
We have used quantum-mechanical calculations in order to find relations between the I 1 SFEs in Mg alloys and fundamental atomic properties of elemental solutes. Our analysis showed that the I 1 SFE is (i) strongly anticorrelated to the atomic volume (Pearson correlation coefficient r = -0.97), (ii) correlated to the electronegativity (r = 0.95) and (iii) logarithmically related to the bulk modulus (r = 0.90). Based on these relations, we propose an YSI that combines all three criteria and rationalizes the complexity of prediction of the I 1 SFEs. As this new numerical indicator reproduces the trends of quantum-mechanically computed SFEs for X Mg 15 with 18 different solutes, we evaluate it for 76 elements from the periodic table of the elements. We have found only one non-RE element, lithium, that we predict to reduce the I 1 SFE. Seeing this limited options in the case of binary Mg alloys, we continue in using the newly introduced YSI parameter in high-throughput theoryguided rapid prototyping of new ternary Mg-alloys with increased ductility.
We extend our YSI analysis to ternary Mg alloys assuming that the physical properties of two solutes can be represented by average values of their individual materials parameters (atomic volume, the electronegativity and the bulk modulus). Out of 2850 solute combinations computed, 133 solute pairs (not including any RE elements) possess a YSI value larger than 0.85 and 11 of them have YSIs higher than 0.95. Subsequent quantummechanical calculations of these 11 solute combinations resulted in lower I 1 SFEs (compared with elemental Mg) in case of all of them. Our proposed YSI parameter thus proved to be 100% effective in identifying promising solute candidates. Importantly, the newly introduced numerical quantifier allows the identification of the most promising solutes in a very fast manner and without computationally expensive and lengthy ab initio calculations.